I am searching for $(KXK)^+ =\left(\begin{array}{cc}X_{N-1}& 0 \\ 0 & 0\end{array} \right)^+$, where $X^{-1}\in\mathbb{R}^N$ exists and $K=\left(\begin{array}{cc}1_{N-1}& 0 \\ 0 & 0\end{array} \right)$.
2026-03-27 01:00:02.1774573202
Pseudoinverse of $KXK$, $K=\left(\begin{array}{cc}1_{N-1}& 0 \\ 0 & 0\end{array} \right)$Pseudoinverse of $KXK$, K=
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$F^+:=\left(\begin{array}{cc}X_{N-1} ^+& 0 \\ 0 & 0\end{array} \right)$ is the unique Pseudoinverse of $F:=KXK$. This holds as $$F^+ FF^+=\left(\begin{array}{cc}X_{N-1} ^+ X_{N-1} X_{N-1} ^+& 0 \\ 0 & 0\end{array} \right)=F^+\\ F F^+F=\left(\begin{array}{cc}X_{N-1} X_{N-1}^+ X_{N-1}& 0 \\ 0 & 0\end{array} \right)=F \\ FF^+, F^+F \text{ are symmetric.} $$